# Why are linear functions linear?

I always thought linear functions need to satisfy $$f(x+y)=f(x)+f(y).$$ I am a tad confused now, consider $f(x)=2x+3$. $f(1)=5$, $f(2)=7$, $f(1+2)=f(3)=9 \neq f(1)+f(2)$ which was what I thought linear functions should satisfy.

Could someone clarify?

• You are right, linearity means $f(x+y)=f(x)+f(y)$. Functions on ${\mathbb R}$ which really are linear are of the form $f(x)=kx$ for some $k\in {\mathbb R}$. THose of form $f(x)kx+n$ with a nonzero $n$ should be called affine. Feb 7, 2015 at 12:12
• The Wikipedia page on "linear function" could have warned you that these are false friends. Feb 7, 2015 at 12:18
• This looks related to Why Rotations (in CG) are not linear?
– robjohn
Feb 7, 2015 at 14:35
• @RespawnedFluff: I wasn't saying it was a duplicate, just related.
– robjohn
Feb 7, 2015 at 17:34
• Feb 8, 2015 at 11:43

You're confusing between two different notions.

In calculus, a linear function is a polynomial function of the form $f(x)=ax+b$.

In linear algebra and functional analysis, a linear function is a linear map. (one of the properties that it satisfies is $f(x+y)=f(x)+f(y)$, known as additivity)

The difference between the two is that the latter needs to have $f(0)=0$. Proof: $$f(0)=f(0+0)=f(0)+f(0)=2f(0)\iff f(0)=0.$$ I discuss this in more detail in my (not yet finished) note.

• actually, $f(x) = a x + b$ is called affine in the general, and linear iff $b=0$ Feb 7, 2015 at 21:19
• @TemplateRex In most standard calculus textbook, a linear function is defined as a function whose graph is a line. That's why I emphasized the distinction between a linear function in the context of calculus, and in the context of linear algebra and functional analysis. Although I do agree that calling it affine would have been much better. Feb 7, 2015 at 21:26
• Not only in calculus. If you look up the definition of a splitting field, it will probably talk about a polynomial splitting "into linear factors". The linear factors are not necessarily of the form $ax$.
– bof
Feb 9, 2015 at 8:20

$f(x)$ = $2x$ + $3$ isn't a linear function (from and to the set of real numbers). You can easily see that $f(x+y)$ = $2x$ + $2y$ + $3$, and that $f(x)$ + $f(y)$ = $2x$ + $2y$ + $6$. Equality obviously fails.

A linear function (as a mapping from and to the set of real numbers) should be in the from $ax$, where $a$ is a constant real number.

• You will find a large number of textbooks and websites which would describe $f(x)=2x+3$ as a "linear function". Others may not, but that merely means that the term is ambiguous. Feb 8, 2015 at 11:36